发表机构
Technion – Israel Institute of Technology; Eindhoven University of Technology（以色列理工学院; 埃因霍温理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对离散时间Lur'e反馈系统，利用循环矩阵的奇异值定义扇区界，提出代数框架判定非零周期不动点的存在性，并给出振幅上下界，尤其适用于继电器反馈系统。
AI 中文摘要
我们研究离散时间Lur'e反馈系统中识别非平凡（即非零）周期不动点的问题。利用由线性子系统（无论稳定或不稳定）的传递函数构造的循环矩阵，我们引入了一个代数框架，该框架使我们能够确定此类不动点何时存在。该框架产生了一个由两个向量定义的扇区界，其斜率对应于循环矩阵的最大和最小正奇异值。假设非线性反馈函数是无记忆的，我们证明了非平凡$P$-周期不动点存在的必要条件是：非线性反馈函数的连续完备化与该扇区界的交集包含至少一个除原点以外的点。我们的刻画为所有周期$P$提供了一个统一的条件，并进一步使我们能够推导出具有有界反馈函数的可容许周期不动点振幅的上界。特别地，对于具有无源反馈函数的继电器反馈系统，我们推导了此类周期不动点振幅的上界和下界。
英文摘要
We study the problem of identifying nontrivial, i.e., nonzero, periodic fixed-points in discrete-time Lur'e feedback systems. Using the circulant matrix constructed from the transfer function of the linear subsystem, whether stable or unstable, we introduce an algebraic framework that allows us to determine when such fixed-points exist. This framework yields a sector bound defined by two vectors, whose slopes correspond to the maximum and minimum positive singular values of the circulant matrix. Assuming that the nonlinear feedback function is memoryless, we show that a necessary condition for the existence of nontrivial $P$-periodic fixed-points is that the intersection of the continuous completion of the nonlinear feedback function with that sector bound contains at least one point other than the origin. Our characterization provides a unified condition valid for all periods $P$, and further enables us to derive upper bounds on the amplitudes of admissible periodic fixed-points with bounded feedback functions. In particular, for relay feedback systems with passive feedback functions, we derive both upper and lower bounds for the amplitudes of such periodic fixed-points.
